For a compact subset X ⊂ R n and a continuous function f : X → C , f is integrable on X ( as both X and f will be bounded). Does this generalise? That is, if X is a compact measure space and f : X → C is a continuous function, under which circumstances (i.e. under which assumptions on the topology of X and under which assumptions on the measure on X) is it true that f is integrable on X? Thanks for any help!

Question

For a compact subset   X  ⊂            R        n   and a continuous function   f  :  X  →      C  , f is integrable on   X ( as both   X and   f will be bounded). Does this generalise? That is, if   X is a compact measure space and   f  :  X  →      C   is a continuous function, under which circumstances (i.e. under which assumptions on the topology of   X and under which assumptions on the measure on   X) is it true that   f is integrable on   X? Thanks for any help!

candelo6a · Accepted Answer

It works if   X is compact and the measure of   X is finite since then      ∫    X        |    f      |    d  μ  ≤      ∫    X    max  (      |    f      |    )  d  μ  =  max  (      |    f      |    )  μ  (  X  )  .

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